Problem 14 - Algebra A Combined Function

Question

Solve each equation and check each proposed solution. See Examples 4 through 6. $$ \frac{6}{4-3 x}=-3 $$

Step-by-Step Solution

Answer

The solution is \(x = 2\).

Step 1: Cross-Multiply to Eliminate the Fraction

Given equation: \( \frac{6}{4-3x} = -3 \). To eliminate the fraction, multiply both sides by \( 4 - 3x \):\[6 = -3(4 - 3x)\]

Step 2: Distribute the Right Side

Distribute \(-3\) on the right side:\[6 = -12 + 9x\]

Step 3: Isolate the Variable Term

Add 12 to both sides to move the constant term:\[6 + 12 = 9x\]Simplify:\[18 = 9x\]

Step 4: Solve for x

Divide both sides by 9 to solve for \(x\):\[x = \frac{18}{9} = 2\]

Step 5: Check the Proposed Solution

Substitute \(x = 2\) back into the original equation to check:\[\frac{6}{4 - 3(2)} = \frac{6}{4 - 6} = \frac{6}{-2} = -3\]The original equation holds with \(x = 2\), confirming the solution.

Key Concepts

Cross-Multiplication

In algebra, cross-multiplication is a technique used to eliminate fractions in an equation, making it easier to solve. Imagine you have a fraction that is causing some complexity in your equation. Cross-multiplication allows you to "get rid of" that fraction in one smooth move. Here's how it works:

• You multiply the numerator of one fraction by the denominator of the other fraction.
• Then, do the same with the remaining numerator and denominator pair.

In our original equation, \(\frac{6}{4-3x} = -3\), you want to eliminate the fraction \(\frac{6}{4-3x}\). To do this, multiply both sides by \(4-3x\): \[6 = -3(4-3x)\] This action "crosses" away the fraction because multiplying by its own denominator effectively cancels the two out. It's like simplifying a complex equation into a much more straightforward one that's easier to handle.

Solving Rational Equations

Solving rational equations is a common practice in algebra. It involves working with equations that have one or more fractions. Here is how you can solve such equations step-by-step:

• Start by eliminating the fractions: Use cross-multiplication to get rid of fractions as shown in the previous section.
• Distribute terms: If any multiplication is needed after cross-multiplication, distribute the terms across the parentheses. For instance, in our example, you distribute the \(-3\) across \(4 - 3x\), resulting in \(-12 + 9x\).
• Isolate the variable: Move terms around to get the variable on one side of the equation and keep constant terms on the other. You add \(12\) to both sides to get: \[6 + 12 = 9x\]
• Solve for the variable: Finally, get the variable itself by performing necessary operations, like dividing both sides by a constant (!\[x = \frac{18}{9} = 2\].

Each step simplifies the equation further until you arrive at the solution for the variable. Remember, the goal is to isolate the variable and keep things on balance.

Checking Solutions

Once you find a solution to your equation, it's crucial to check that your solution actually works in the original equation. This step ensures that no mistakes have been made and reinforces your understanding of the algebraic process.Here's how we check our solution in the exercise:

• Substitute the solution back into the original equation: Replace the variable with the value you found. In this case, substitute \(x = 2\) into the original equation \(\frac{6}{4-3x} = -3\).
• Perform the arithmetic operations: Calculate each side of the equation separately using your substituted value.\[\frac{6}{4 - 3(2)} = \frac{6}{4 - 6} = \frac{6}{-2} = -3\]
• Confirm both sides are equal: Check if your calculations make the left side equal to the right side of your original equation.

This process confirms \(x = 2\) is indeed a valid solution because substituting it into the equation results in a true statement: \(-3 = -3\). Double-checking your solution is especially important as it verifies correctness, preventing potential errors in further calculations.